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LINEAR σADDITIVITY AND SOME APPLICATIONS
, 906
"... Abstract. We show that countable increasing unions preserve a large family of wellstudied covering properties, which are not necessarily σadditive. Using this, together with infinitecombinatorial methods and simple forcing theoretic methods, we explain several phenomena, settle problems of Just, ..."
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Abstract. We show that countable increasing unions preserve a large family of wellstudied covering properties, which are not necessarily σadditive. Using this, together with infinitecombinatorial methods and simple forcing theoretic methods, we explain several phenomena, settle problems of Just, Miller, Scheepers and Szeptycki [15], Gruenhage and Szeptycki [13], Tsaban and Zdomskyy [33], and Tsaban [29, 32], and construct topological groups with very strong combinatorial properties. 1.
A unified theory of function spaces and hyperspaces: local properties, submitted
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FRÉCHETURYSOHN FOR FINITE SETS, II
, 2006
"... Abstract. We continue our study [6] of several variants of the property of the title. We answer a question in [6] by showing that a space defined in a natural way from a certain Hausdorff gap is a Fréchet α2 space which is not FréchetUrysohn for 2point sets (FU2), and answer a question of Hrusak ..."
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Abstract. We continue our study [6] of several variants of the property of the title. We answer a question in [6] by showing that a space defined in a natural way from a certain Hausdorff gap is a Fréchet α2 space which is not FréchetUrysohn for 2point sets (FU2), and answer a question of Hrusak by showing that under MAω1, no such “gap space ” is FU2. We also introduce versions of the properties which are defined in terms of “selection principles”, give examples when possible showing that the properties are distinct, and discuss relationships of these properties to convergence in product spaces, to the αispaces of A.V. Arhangel’skii, and to topological games. 1.
On λ′sets
, 2002
"... A set X ⊆ 2 ω is a λ′set iff for every countable set Y ⊆ 2 ω there exists a Gδ set G such that (X ∪ Y) ∩ G = Y. In this paper we prove two forcing results about λ′sets. First we show that it is consistent that every λ′set is a γset. Secondly we show that is independent whether or not every (†) ..."
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A set X ⊆ 2 ω is a λ′set iff for every countable set Y ⊆ 2 ω there exists a Gδ set G such that (X ∪ Y) ∩ G = Y. In this paper we prove two forcing results about λ′sets. First we show that it is consistent that every λ′set is a γset. Secondly we show that is independent whether or not every (†)λ′set is a λ′set.
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"... A countable FréchetUrysohn space of uncountable character. (English summary) Topology Appl. 155 (2008), no. 10, 1129–1139. Recall that a collection A of subsets of the natural numbers ω is an almost disjoint family if each A in A is infinite, and for two different elements A, B ∈ A, A ∩ B  < ℵ ..."
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A countable FréchetUrysohn space of uncountable character. (English summary) Topology Appl. 155 (2008), no. 10, 1129–1139. Recall that a collection A of subsets of the natural numbers ω is an almost disjoint family if each A in A is infinite, and for two different elements A, B ∈ A, A ∩ B  < ℵ0. Given an almost disjoint family A on ω, the MrówkaIsbell space Ψ(A) is defined as follows: the underlying set is ω ∪ A; every point n ∈ ω is isolated; an open canonical neighborhood of A ∈ A is of the form {A} ∪ (A � F), where F ⊆ ω is finite. It is well known that Ψ(A) is zerodimensional, Hausdorff and locally compact. Let X(A) be the subspace of the onepoint compactification Ψ(A) ∪ {∞} of Ψ(A), whose underlying set is ω ∪ {∞}. In the paper under review, the author introduces the notion of strong completely separable almost disjoint families and then uses it to show the existence of an infinite almost disjoint family B such that X(B) n is FréchetUrysohn for every n ∈ ω, and
Countable Fréchet Boolean groups: An independence result
, 2008
"... It is relatively consistent with ZFC that every countable FUfin space of weight ℵ1 is metrizable. This provides a partial answer to a question of G. Gruenhage and P. Szeptycki [GS1]. ..."
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It is relatively consistent with ZFC that every countable FUfin space of weight ℵ1 is metrizable. This provides a partial answer to a question of G. Gruenhage and P. Szeptycki [GS1].
Countable Fréchet Boolean groups: An independence result
, 2008
"... It is relatively consistent with ZFC that every countable FUfin space of weight ℵ1 is metrizable. This provides a partial answer to a question of G. Gruenhage and P. Szeptycki [GS1]. ..."
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It is relatively consistent with ZFC that every countable FUfin space of weight ℵ1 is metrizable. This provides a partial answer to a question of G. Gruenhage and P. Szeptycki [GS1].